Many beam-to-column connections, consisting of rectangular hollow section (RHS) columns and wide flangeI-beam connections, sustain brittle fracture of welded connections at the beam ends during a largeearthquake. These fractures most frequently occur in regions around the beam bottom flange groovewelds. A series of tests was conducted on an improved RHS column-to-I beam connection. The aim of thetests was to find possible solutions for avoiding premature occurrences of brittle fracture in RHScolumn-to-I beam connections. Research results show that the improved connection does not fail by fractureas observed in the conventional connections and has a larger energy dissipation capacity than the conven-tional types. This paper describes the finite element modeling method employed to analyze the new RHScolumn-to-I beam connection. The ABAQUS finite element package is used to simulate the experimental be-havior, and three highly detailed 3D finite element models are created. These are complex models accountingfor material nonlinearity, large deformation and contact behavior. The connection models have beenanalyzed through the elastic and plastic ranges up to failure. Comparisons with experimental data showthat the models have high levels of accuracy.
The 1995 Kobe earthquake revealed that conventional types ofrectangular hollow section (RHS) column-to-I beam connections insteel building frames were vulnerable to brittle fracture under strongground motion. During the earthquake, most cracks in the RHScolumn-to-I beam connections started with ductile tear, and thenchanged to brittle fracture at the beam-ends after the connectionssustained extensive yielding or local buckling. After the earthquake,improvements of connection design were proposed. One way toachieve sufficient strength of connections is to reinforce the beamends by adding additional members or plates [1].
This strengthening approach was developed at Kumamoto Univer-sity in Japan [1]. A new connection detail is to use the through-diaphragm in which a cut is prepared along the edge of thethrough-diaphragm so that the beam flange fits into the cut, creatinga welded joint of rabbet (U-shape) to reinforce the beam end strength.Experimental tests were conducted in 1999 [3]. After a series of tests onboth conventional and improved connections, it was found thatfractures in connections did not occur in the improved connectionand also a larger energy dissipation capacity was achieved. These testresults were reported in [3].
l rights reserved.
Experimental tests in general can provide reliable results that candescribe the behavior of the beam-to-column connection. However,experimental work involves high costs, and in some cases, is not fea-sible. Since numerical methods offer more flexibility and the possibil-ity to investigate a wider range of parameters than experiments cancover, the research on beam-to-column connections is further ex-tended using the finite element method.
The main objective of this work is to validate the finite elementmethod by comparing with the experimental results. This paperfirst reviews the experimental procedure conducted in [3] and failuremodes of the connections, and then, describes the finite elementmodeling method that is used to analyze the connections. TheABAQUS finite element package is used to simulate the new RHScolumn-to-I beam connection behavior observed in the testsperformed at Kumamoto University in Japan. The results obtainedfrom the finite element analyses are evaluated by comparing the mo-ment–rotation responses with those of the corresponding tests.
2. Specimen and test program [3]
2.1. Specimen design
In order to compare different test results, specimens are made intwo types: T-1, and T-2 and T-3, as shown in Fig. 1. T-1 is a conven-tional connection type, with ENDMu ≥ 1.3Mp, where ENDMu is the
43J. Wu, Y.T. Feng / Journal of Constructional Steel Research 86 (2013) 42–53
ultimate flexural capacity of the specimen and Mp is the full plasticmoment of the beam; T-2 and T-3 are the improved types, withENDMu = α(fMu + wMp), where fMu is the ultimate moment of thebeam flange, and wMp is the plastic moment of the beam web, and
α ¼ Lp þ Lh þ LtLh þ Lt
in which the lengths Lh, Lp and Lt are shown in Fig. 1. Note that inorder to compare the deformation capacity, Lp in T-3 is 65 mm longerthan in T-1, and 40 mm longer than in T-2.
Fig. 2. Connecti
2.2. Specimen details
Fig. 2 illustrates the connection details for the conventional and im-proved connections of specimen. In each of these specimens, a wideflange beam with nominal dimensions of 500 mm × 200 mm (beamheight × flange width) is welded to a cold-formed square hollow sec-tion column with nominal dimensions of 400 mm × 400 mm (columnwidth × depth) to form a T-shaped subassembly. In specimen T-1, thebeam web is welded to the column flange, and both beam flanges arewelded to the through-diaphragm plates, and in specimens T-2 andT-3, both beam flanges are welded to the through-diaphragm plateswhile the beam web is bolted to the fin plate by a single row of five
Note: σy = yield stress; σu = ultimate tensile strength stress; E.L. = elongation; E =elastic modulus; Y.R. = yield ratio (σy/σu); t = thickness.
44 J. Wu, Y.T. Feng / Journal of Constructional Steel Research 86 (2013) 42–53
bolts. The column and beam have a length of 3.5 m and 2.4 m respec-tively. Both ends of the column are fully fixed.
2.3. Material properties
All the tensile uniaxial tests are done to the samples taken fromone section of the beam and column. The mechanical properties ofthe materials are summarized in Table 1. All the materials are ofweldable low-carbon steel as specified by the Japanese IndustrialStandards (JIS) SS400 and STKR400, roughly equivalent to the USASTM A36 and A501 steel standards.
2.4. Test set-up
The loading arrangements are shown in Fig. 3. Displacementstaken from measurements are not only the horizontal displacementu1 at the loading point but also the vertical displacements v1 and v2at the diaphragms and the horizontal displacement u2 at the columnend. The rotation of beam, θm, is calculated by the following equation:
θm ¼ u1−u2
L− v2−v1
Jd
Fig. 3. Test
where L and Jd respectively denote the distance from the loadingpoint to the column face and the distance between the centroids ofthe top and bottom beam flange.
All the specimens are subjected to a slowly applied cyclic loadingand are tested as follows: at least 2 cycles of reversed loading in anelastic region and, subsequently, a displacement controlled cyclicloading with the amplitude increased as ±2θp, ±4θp, ±6θp,…, untila failure occurs, where θp represents the elastic beam rotation at thefull plastic moment Mp (see Appendix 2) [1,2], and can be calculatedby the following equation:
θp ¼ L3EI
Mp þ1
GAw
Mp
L:
Two cycles of the loading are applied at each displacementincrement.
2.5. Test results
Fig. 4 shows the bending moment vs. beam rotation hysteric curveat the column face. The moment is the maximum moment at the col-umn face Mm and is non-dimensionalized by the full-plastic momentMp of the beam. The moment takes a positive value when the bottomflange is in tension. The rotation θm denotes the rotation of the beamsegment between the loading point and the column face (seeAppendix 2). Tables 2–4 show the failure sequences of specimens.
One of the important failure modes observed in the test is tensilefailure of the beam flanges. First, cracks are found either at the tip ofthe weld toes or at the toe of the beam cope. Then, these cracks growgradually with load cycling and finally lead to tensile failure of thebeam flange in T-1 and T-2 (see Fig. 5). T-3, which is designed tohave sufficient tensile capacity at the flange ends, reaches the maxi-mum loads, but only leads to local buckling of the top and bottomflanges and the web of the beams (see Fig. 6).
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3. Finite element analyses
3.1. Finite element model
A three-dimensional finite element model of the RHS column-to Ibeam connection is created using the HYPERMESH (Altair) pre-processor. The connections are analyzed using the ABAQUS finite ele-ment software package.
The finite element mesh used in the analyses is shown in Fig. 7.The mesh contains approximately 36,481 nodes and 25,569 elements.In the mesh generation, the connection is divided into six individualcomponents, which are shown in Fig. 8, and are referred to as the col-umn, beam, fin plate, diaphragm, bolt, and weld.
1) Column, beam, diaphragm and fin plateThe column, beam, diaphragm and fin plate are modeled using 3D8-node brick elements. Linear hybrid elements are selected to pre-vent the possible problem of volume strain locking. Following aconvergence study, four elements through the thickness of the di-aphragm and beam flange are used. Furthermore, to reduce thenumber of elements and nodes in the FE model, only one elementis used for the column, beam web and fin plate through theirthickness.
Fig. 5. Tensile failure in T-1 and T-2.
Fig. 6. Local buckling observed in T-3.
Fig. 8. Mode components.
46 J. Wu, Y.T. Feng / Journal of Constructional Steel Research 86 (2013) 42–53
2) WeldThere are two types of welding. One type is the filled weld, appliedto the fin plate-to-column connection. The other is groove weld,applied to the diaphragm-to-column and beam-to-diaphragmconnections. In the finite element model, the weld is assumed tobe an extension of the diaphragm sections, thus the column sec-tions and the fin plate section have the same material propertiesas the connection. Consequently, the weld is modeled as an indi-vidual component in the connection model using 8-node brickelements.
3) BoltThe bolt model is constructed using 8-node brick elements and6-node triangular prism elements, as shown in Fig. 9. The bolt con-nections include three main parts: beam, fin plate and bolts. Tocapture the accurate stress behavior, an intensive mapped meshis made around the boltholes. The hexagon bolt heads aremodeled as cylinders.
A small sliding interaction behavior between the contacting sur-faces is considered for all the contact surfaces in order to fully transferthe loading from the beam web to the fin plate, and finally, to thecolumn. The following contact interactions for the bolt connectionare considered: 1) contact between the bolt shank and the bolt
Fig. 7. Finite element mesh.
hole; 2) contact between the bolt head and the beam web; 3) contactbetween the nut and the fin plate; and 4) contact between the finplate and the beamweb. Pre-tension is not applied to the bolts duringthe analysis. A friction coefficient μ = 0.45 is used for all the contactsurfaces.
Although two cycles of loading are applied at each displacementincrement in the experiments, in the numerical analyses, only onecycle for each displacement increment is exerted to the beam tosave computational time.
3.2. Classical metal plasticity model
In this work, the classical metal plasticity constitutive modelimplemented in ABAQUS is adopted for its simplicity. The total strainrate is decomposed into the elastic and plastic components:
_ε ¼ _εel þ _εpl ð1Þ
where εel and εpl are elastic and plastic strain, respectively. Eq. (1)can be written in integrated form as
ε ¼ εel þ εpl: ð2Þ
Fig. 9. Finite element model of bolt.
Fig. 10. Steel stress–strain curve.
47J. Wu, Y.T. Feng / Journal of Constructional Steel Research 86 (2013) 42–53
The elasticity is linear and isotropic, and is written in terms of thebulk modulus, K, and the shear modulus, G, as follows
p ¼ −Kεv; S ¼ 2Geel ð3Þ
with
K ¼ E3 1−2νð Þ ; G ¼ E
2 1þ νð Þ ð4Þ
where p ¼ −1=3 trace σð Þ is the equivalent pressure stress; εv =trace(ε) is the volume strain; S ¼ σþ pI is the deviatoric stress; andeel = εel − 1/3εvI is the elastic deviatoric strain.
The von Mises yield criterion with isotropic hardening behavior isassumed:
f σð Þ ¼ffiffiffiffiffiffiffiffiffiffiffiffiffi32S : S
r¼ σ0 εpl
� �ð5Þ
where σ0 is the yield stress and is defined as a function of the equiv-alent plastic strain εpl. It can be determined according to the stress–strain relation measured from uniaxial tensile experiments, asshown in Fig. 10.
The associated flow rule is used in the model and has the follow-ing form
dεpl ¼ dεpln ð6Þ
-150
-100
-50
0
50
100
150
M ×
10
(KN
.m)
θ
Experiment
FE
-0.04 -0.03 -0.02 -0.01
Fig. 11. Hysteresis
with
n ¼ 32
Sf σð Þ : ð7Þ
Eqs. (2) to (7) define the metal material behavior. These equationsare required to be integrated and solved when plastic flow occurs. Theintegration is done by applying the backward Euler method to the flowrule, and then a nonlinear equation is established by combining theequationswith the integrated flow rule. The nonlinear equation is solvedby Newton's method and the metal plasticity behavior is determined.
For the ABAQUS simulation, the nominal engineering stress–strainrelation obtained from the steel tensile coupon test (see Table 1) isconverted to the true stress–strain relationship, according to: σtru ¼σnom 1þ εnomð Þ and εtru = ln(1 + εnom), in which σtru and εtru repre-sent the true stress and strain, respectively, andσnom and εnom are thenominal stress and strain respectively.
3.3. Analysis results
The numerical analyses of specimens are terminated when theconnections fail with the maximal rotation 6θp. Figs. 11–13 showboth the simulated and experimental results, and reveal that whenthe beam tip rotation is between 4θp and −4θp, the experimentaland numerical loops match well. However, when the rotation isover 4θp and −4θp, the numerical loops overestimate the stiffnessof the specimens. The difference between the experimental and
(Rad)
0 0.01 0.02 0.03 0.04
curves (T-1).
-150
-100
-50
0
50
100
150
M ×
10
(KN
.m)
θ (Rad)
Experiment
FE
0-0.04 -0.03 -0.02 -0.01 0.01 0.02 0.03 0.04
Fig. 12. Hysteresis curves (T-2).
48 J. Wu, Y.T. Feng / Journal of Constructional Steel Research 86 (2013) 42–53
numerical loops is likely caused by the difference between the nomi-nal andmeasured material properties and inaccurate measurement ofthe specimen dimensions.
The moment–rotation relationships for specimens T-1, T-2 andT-3 analyzed by FEM are shown in Fig. 14, in which the curves, test1-1 (FE), test 2-1 (FE), and test 3-1 (FE), are obtained from the FEanalyses, and test 1, test 2, and test 3 are obtained from theso-called skeleton curves. The skeleton curve is constructed from ahysteretic curve by linking a portion of the curve that exceeds themaximum load in the preceding loading cycle sequentially (seeAppendix 3). Comparison of the FE results with the experimental mo-ment–rotation curve (see Figs. 15–17) shows good agreement interms of general behavior and the maximum values with a differenceof around 10%. An observation of FE failure mode shows a similarity tothe test results (see Figs. 5 and 18).
4. Prediction of ultimate strength of connections
4.1. Failure modes
Two failure modes are identified in the connection test: 1) the localbuckling of the plate element at the beam end and 2) the tensile failure
-150
-100
-50
0
50
100
150
M ×
10
(KN
.m)
θ
Experiment
FE
-0.05 -0.03 -0.01
Fig. 13. Hysteresis
of the beam flange at the beam end. The fracture paths are shown inFig. 19.
The ultimate strength of these connections can be predicted bysimple formulas based on an elementary plastic analysis.
4.2. Tensile strength of the welded joints
To predict the ultimate tensile strength of the welded joints, twofracture paths are proposed as follows (see Fig. 20). Lp signifies thelength of the welded joint, Ld signifies the space between the toe ofthe beam cope and the beam end, Ls signifies the space between thebeam end and column face, b denotes the width of the beam flangeand be denotes the width of the welded joint.
N. Tanaka [5] recommended the following formula for calculationof the tensile strength of each fracture path when failure occurs byductile tear along the fracture path (fracture path 1),
Pu ¼ Lf � tf � sinθþ cosθffiffiffi3
p� �
� σ f ;u ð8Þ
where Lf signifies the length of the fracture path (see Fig. 19), σf,u isthe ultimate tensile strength of the beam flange, and tf is the thick-ness of the beam flange.
(Rad)
0.01 0.03 0.05
curves (T-3).
100
90
80
70
60
test 1 test 2
test 3test 2-1 (FE)
test 1-1 (FE)test 3-1 (FE)
50
40
30
20
10
00 0.02 0.04 0.06
M ×
10
(KN
.m)
θ (Rad)
Fig. 14. Moment–rotation relationships.
90
80
70
60
50
40
30
20
10
00 0.020.01 0.04 0.050.03 0.06
M ×
10
(KN
.m)
θ (Rad)
test 1
test 1-1 (FE)
Fig. 15. T-1 moment–rotation relationships.
90
80
70
60
50
40
30
20
10
00 0.02
test 2
test 2-1 (FE)
0.04 0.06
M ×
10
(KN
.m)
θ (Rad)
Fig. 16. T-2 moment–rotation relationships.
49J. Wu, Y.T. Feng / Journal of Constructional Steel Research 86 (2013) 42–53
90
100
80
70
60
50
40
30
20
10
00 0.02
test 3
test 3-1 (FE)
0.04 0.06
M ×
10
(KN
.m)
θ (Rad)
Fig. 17. T-3 moment–rotation relationships.
50 J. Wu, Y.T. Feng / Journal of Constructional Steel Research 86 (2013) 42–53
The ultimate moment carried by the flange Mf,u is evaluated as theaxial capacity of one of the beam flange-to-column flange joints mul-tiplied by the distance between the centroids of the top and bottomflange to column joints. Therefore, the flexural capacity of the beamat the column face Mf,u is given as
Mf ;u ¼ b� tf � σ f ;u þ2ffiffiffi3
p Lp−Ld� �
� tf � σ f ;u
� �� Jb � L
Lh þ Ltð Þ : ð9Þ
In general, fracture path 2 is much stronger than fracture path 1and thus such a fracture is not necessary to be considered. However,to ensure a sufficient over-strength to prevent tensile failure, the op-timum length of such welded joint can be calculated as
Lp ¼ Ld þffiffiffi3
p
2be−bð Þ: ð10Þ
Fig. 18. T-1 plastic strain distribution.
The flexural capacity of bolted web connection in specimens T-2and T-3 can be neglected because of the large flexibility of theconnection due to the bolt slip and local yielding of the column flange.However, specimen T-1 has a welded web joint, and the bending mo-ment carried by this joint is added to the moment given by Eq. (9) toevaluate the ultimate flexural capacity of the connection. AIJ [4]proposed the following formula for the flexural capacity,
Mw;u ¼ tw x−Svð Þ Hb−tf−x−Svð Þσw;y ð11Þ
where x is obtained by a yield line analysis and is given by
x ¼ 4 Hb−tfð ÞM0
fwþ S2v
� �1=2ð12Þ
M0 ¼ t2cσc;y
4
Fig. 19. Fracture paths at the beam end in tests.
Fig. 20. Proposed fracture paths at the beam end.
Table 5The tensile capacity comparison between the test and prediction results.
Specimen Test Prediction Mm/EndMu
Mm (kN m) Mf,u (kN m) Mw,u (kN m) EndMu (kN m)
T-1 830 781 55 836 0.99T-2 830 854 854 0.97
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where Hb denotes the height of the beam, tc is the thickness of thecolumn, tw is the thickness of the beam web, Sv is the vertical spaceof the beam cope (see Fig. 21), and fw signifies the stress and isgiven as the smaller of the following two values,
fw ¼ twσw;y; fw ¼ 2ffiffiffi3
p tcσc;y ð13Þ
Fig. 21. Moment carried b
where σw,y is the yield strength of the beam web, and σc,y is the yieldstrength of the column.
The ultimate flexural capacities EndMu = Mf,u or EndMu = Mf,u +Mw,u of beam-to-column connections, determined by the tensile ca-pacities of the welded joints at the beam ends, are summarized inTable 5 and compared with the test results. The table shows thatthe predictions agree well with the test results. Note that specimenT-3 was terminated by local buckling. Although local buckling ofplate elements is governed by another formula, Eqs. (9) and 10 arealso suitable for evaluation of the ultimate strength of the connection.A more detailed explanation is omitted here.
5. Conclusion
A FE analysis of the behavior of the new RHS column-to-I beam con-nections is described in this paper. Three test specimens are simulated.The model includes the individual beams, columns, diaphragms, bolts,fin plate and the complex contact surfaces. Material nonlinearity is con-sidered for all components. Contacts are critical to model the boltedconnection behavior of the joint. Contact elements have been used atthe bolt-hole and also at the surface between the web of the beamand fin plate, taking into consideration the friction between the sur-faces. Three-dimensional brick elements are employed as this type of
y welded web joint.
52 J. Wu, Y.T. Feng / Journal of Constructional Steel Research 86 (2013) 42–53
element is easily adapted to model interfaces between the connectingsurfaces. The comparison shows a good correlation between the FEand experimental results of the connection behavior. This proves thatthe FEM is capable of accurately predicting RHS column-to-I beamconnection behavior. A new type of RHS column-to-I beam connectionmodel has successfully been created and validated against a series oftest data. The methodology can now be employed in parametric studyto develop more new connections, for example, to change the shapesof welded joint at the beam end. The results will be reported later.
Appendix 1.Symbols
A cross-sectional area (mm2)E elastic modulus (Gpa)
a
σ strength (N/mm2)G shear modulus of elasticity (Gpa)H height (mm)I moment of inertia (mm4)J distance between centroids (mm)L length (mm)M bending moment (N·mm or kN·mm)P force (N or kN)T thickness (mm)u horizontal displacement (mm)v vertical displacement (mm)θ rotation (radian)
Appendix 2. Definition of Mp and θp
The bending moment Mm denotes the maximum beam moment at the column face. The rotation θm denotes the rotation of the beamsegment between the loading point and the column face (see Fig. a). The full plastic moment Mp is calculated by using the measured yield stressesof the materials and measured dimensions of the beam sections. The elastic beam rotation θp at the full plastic moment is defined as the elastic com-ponent of beam rotation at Mm = Mp (see Fig. b).
b
Appendix 3. Definition of skeleton curve
The skeleton curve is constructed from a hysteretic curve by linking a portion of the curve that exceeds the maximum load in the precedingloading cycle sequentially (see Fig. c).
c
53J. Wu, Y.T. Feng / Journal of Constructional Steel Research 86 (2013) 42–53
References
[1] Kurobane Y. Improvement of I beam-to-RHS column moment connections foravoidance of brittle fracture. In: Choo YS, Van Der Vegte GJ, editors. Tubular struc-tures VIII. Rotterdam: Balkema; 1998. p. 3–17.
[2] Wu Jian, Ikebata Kotaro, Kurobane Yoshiaki, Makino Yuji, Ochi Kenshi, TanakaMasamitsu. Experimental study on RHS column to wide flange I-beam connectionswith external diaphragms. Hiroshima: AIJ; 1999. p. 517–20 [in Japanese].
[3] WU Jian (2000). Study of new RHS column-to-I beam connections for avoiding ten-sile fracture. M.S. thesis, Kumamoto University, Japan (in Japanese).
[4] AIJ. The state of art report on the structural behavior of steel connections. Tokyo,Japan: Architectural Institute of Japan; 1996. p. 29–31 [in Japanese].
[5] Tanaka, N 1999. Study on structural behavior of square steel column to H-shapedsteel beam connections, doctoral dissertation submitted to Kumamoto University,in Japanese.
